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Subnormal subgroups of En(R) have no free, non-abelian quotients, when n≧3

Published online by Cambridge University Press:  20 January 2009

A. W. Mason
A. W. Mason
Affiliation:
Department of MathematicsUniversity Of Glasgow University GardensGlasgow G12 8QW
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Abstract

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It is known that for certain rings R (for example R = ℤ, the ring of rational integers) the group GL2(R) contains subnormal subgroups which have free, non-abelian quotients. When such a subgroup has finite index it follows that every countable group is embeddable in a quotient of GL2(R). (In this case GL2(R) is said to be SQ-universal.) In this note we prove that the existence of subnormal subgroups of GL2(R) with this property is a phenomenon peculiar to “n = 2”.

For a large class of rings (which includes all commutative rings) it is shown that, for all , no subnormal subgroup of En(R) has a free, non-abelian quotient, when n≧3. (En(R) is the subgroup of GLn(R) generated by the elementary matrices.) In addition it is proved that, if is an SRt-ring, for some t ≧ 2, then no subnormal subgroup of GLn(R) has a free, non-abelian quotient, when n ≧ max (t, 3). From the above these results are best possible since ℤ is an SR3 ring.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 113 - 119
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Bass, H., Algebraic K-theory (Benjamin, New York, Amsterdam, 1968).Google Scholar
2.Grunewald, F. J. and Schwermer, J., Free non-abelian quotients of SL 2 over orders of imaginary quadratic numberfields, J. Algebra 61 (1981), 298–304.CrossRefGoogle Scholar
3.Liehl, B., On the group SL 2 over orders of arithmetic type, J. Reine Angew. Math. 323 (1981), 153–171.Google Scholar
4.Lubotzky, A., Free quotients and the congruence kernel of SL 2, J. Algebra 77 (1982), 411–418.CrossRefGoogle Scholar
5.Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Springer-Verlag, Berlin, 1977).Google Scholar
6.Mason, A. W., A note on subgroups of GL(n, A) which are generated by commutators, J. London Math. Soc. (2) 11 (1975), 509–512.Google Scholar
7.Mason, A. W., Free quotients of congruence subgroups of SL 2 over a Dedekind ring of arithmetic type contained in a function field, Math. Proc. Cambridge. Philos. Soc. 101 (1987), 421–429.Google Scholar
8.Mason, A. W., Free quotients of congruence subgroups of SL 2 over a coordinate ring, Math. Z. 198 (1988), 39–51.Google Scholar
9.Neumann, P. M., The SQ-universality of some finitely presented groups, J. Austral. Math. 16 (1973), 1–6.CrossRefGoogle Scholar
10.Newman, M., Free subgroups and normal subgroups of the modular group, Illinois J. Math. 8 (1964), 262–265.Google Scholar
11.Serre, J-P., Le probleme des groupes de congruence pour SL 2, Ann. of Math. 92 (1970), 489–527.CrossRefGoogle Scholar
12.Suslin, A. A., On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11 (1977), 221–238.Google Scholar
13.Vaserstein, L. N., On the stabilization of the general linear group over a ring, Math. USSR Sb. 8 (1969), 383–4OO.CrossRefGoogle Scholar
14.Vaserstein, L. N., On the normal subgroups of GL n over a ring, in (Lecture Notes in Mathematics, 854, Springer-Verlag, 1981), 456–465.Google Scholar
15.Vaserstein, L. N., Normal subgroups of the general linear groups over von Neumann regular rings, Proc. Amer. Math. Soc. 96 (1986), 209–214.CrossRefGoogle Scholar
16.Vaserstein, L. N., The subnormal structure of general linear groups, Math. Proc. Cambridge Philos. Soc. 99 (1986), 425–431.Google Scholar
17.Vaserstein, L. N., Subnormal structure of the general linear groups over Banach algebras, J. Pure Appl. Algebra 52 (1988), 187–195.CrossRefGoogle Scholar
18.Wehrfritz, B. A. F., Infinite Linear Groups (Springer-Verlag, Berlin, 1973).CrossRefGoogle Scholar